A comparison theorem for semimartingales and its applications. (English) Zbl 0607.60043

Yan, Jia-an

Probabilités XX, Proc. Sémin., Strasbourg 1984/85, Lect. Notes Math. 1204, 349-351 (1986).

[For the entire collection see Zbl 0593.00014.]
Let X be a semimartingale which has a unique decomposition: \(X=X_ 0+M+A\), where M is a continuous local martingale, and A is of finite variation. We denote by \(A^ c\) the continuous part of A and by \(L^ 0(X)\) the local time of X at 0. Under the assumptions that \(L^ 0(X)=0\), \(\Delta\) \(X\leq 0\) and \(\int^{.}_{0}I_{\{X_{s-}>0\}}dA^ c_ s\leq 0\), it is shown that we have \(X\leq 0\) on the set \(\{X_ 0\leq 0\}\). As an application of this result we get a generalization of the comparison theorem for stochastic differential equations given by N. Ikeda and S. Watanabe [Stochastic differential equations and diffusion processes. (1981; Zbl 0495.60005), p. 352].

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44	Martingales with continuous parameter

Keywords:

local martingale; local time; comparison theorem for stochastic differential equations

Citations:

Zbl 0593.00014; Zbl 0495.60005